Procedure: Generate an optimised Latin hypercube design

Description and Background

A Latin hypercube (LHC) is a random set of points in \([0,1]^p\) constructed so that for \(i=1,2,\ldots,p\) the i-th coordinates of the points are spread evenly along [0,1]. However, a single random LHC will rarely have good enough space-filling properties in the whole of \([0,1]^p\) or satisfy other desirable criteria. Therefore, it is usual to generate many LHCs and then select the one having the best value of a suitable criterion.

We present here the procedure for a general optimality criterion and also include details of the most popular criterion, known as maximin. A thorough consideration of optimality criteria may be found in the discussion page on technical issues in training sample design for the core problem (DiscCoreDesign).

Inputs

  • Number of dimensions \(p\)
  • Number of points desired \(n\)
  • Number of LHCs to be generated \(N\)
  • Optimality criterion \(C(D)\)

Outputs

  • Optimised LHC design \(D = \{x_1, x_2, \ldots, x_n\}\)

Procedure

Procedure for a general criterion

  1. For \(k=1,2,\ldots,N\) independently, generate a random LHC \(D_k\) using the procedure page on generating a Latin hypercube (ProcLHC) and evaluate the criterion \(C_k = C(D_k)\).
  2. Let \(K=\arg\max\{D_k\}\) (i.e. \(K\) is the number of the LHC with the highest criterion value).
  3. Set \(D=D_K\).

Note that the second step here assumes that high values of the criterion are desirable. If low values are desirable, then change argmax to argmin.

Maximin criterion

A commonly used criterion is

\[C(D) = \min_{j\ne j^\prime}|x_j - x_{j^\prime}|,\]

where \(|x_j - x_{j^\prime}|\) denotes a measure of distance between the two points \(x_j\) and \(x_{j^\prime}\) in the design. The distance measure is usually taken to be squared Euclidean distance: that is, if \(u=(u_1,u_2,\ldots,u_p)\) then we define

\[|u| = u^T u = \sum_{i=1}^p u_i^2.\]

High values of this criterion are desirable.

Additional Comments

Note that the resulting design will not truly be optimal with respect to the chosen criterion because only a finite number of LHCs will be generated.